Fix $p \in \mathbb{P}^1$. Let $X=\mathbb{P}^1\times \mathbb{P}^1$, $C_1=\mathbb{P}^1\times \{p\}$ and $C_2=\{p\}\times \mathbb{P}^1$. Since $\mathrm{Ext}^1(\mathcal{O}_{C_2},\mathcal{O}_{C_1})\cong \mathbb{C}$, there exists a nontrivial extension $0\rightarrow \mathcal{O}_{C_1} \rightarrow E \rightarrow \mathcal{O}_{C_2} \rightarrow 0$. My question is, how should one understand this sheaf $E$? Is there an intuitive, geometric description of this sheaf $E$?
Edit It turned out that I misunderstood something, and so I changed the latter half of the original question. Thank you for pointing out my confusion.